This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

Transcription

1 This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

2 Available online at European Journal of Operational Research 191 (2008) Short Communication Financial Giffen goods: Examples and counterexamples q Rolf Poulsen a, *, Kourosh Marjani Rasmussen b a Department of Mathematical Sciences at University of Copenhagen, Universitetsparken 5, DK-2100, Denmark b Informatics and Mathematical Modelling, Technical University of Denmark, Bldg. 305, DK-2800, Denmark Received 8 November 2006; accepted 29 June 2007 Available online 13 August Abstract In the basic Markowitz and Merton models, a stock s weight in efficient portfolios goes up if its expected rate of return goes up. Put differently, there are no financial Giffen goods. By an example from mortgage choice we illustrate that for more complicated portfolio problems Giffen effects do occur. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Finance; Portfolio choice; Giffen good; Mortgage planning 1. Introduction A Giffen good is one for which demand goes down if its price goes down. At first, it is counterintuitive that such goods exist at all. But most introductory text-books in economics will tell you that they do; some with stories about potatoes and famine in Ireland, some with first order conditions for constrained optimization. In this note we study similar effects by which we mean a negative relation between expected return and demand in portfolio choice models. Surprising dependence on expected rates of return is not uncommon in finance. In complete models, option prices do not depend on the q We thank Bent Jesper Christensen, Bjarne Astrup Jensen, and Carsten Sørensen for helpful comments. * Corresponding author. Tel.: addresses: rolf@math.ku.dk (R. Poulsen), kmr@imm.dtu.dk (K.M. Rasmussen). stock s growth rate. And quite generally call-option prices increase with the interest rate; immediately you would think that cash-flows are discounted harder, but in fact the replicating strategy which entails a short position in the bank-account becomes more expensive, and hence the call-option does too. We first show that in the basic Markowitz mean/ variance model, there are no Giffen goods; if a stock s expected rate of return goes up, its weight in any efficient portfolio goes up. This seems a text-book comparative statics result. We have, however, only been able to find it indirectly stated, for instance one could view it as a corollary or lemma related to the Harmony Theorem from Luenberger (1998, Section 7.8). So we give a simple proof. We then look at Merton s dynamic investment framework. In its basic version demand for any asset depends positively on its expected rate of return, but if a subsistence level is included, demand for the risk-free asset may fall with the interest rate /$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi: /j.ejor

3 572 R. Poulsen, K.M. Rasmussen / European Journal of Operational Research 191 (2008) Skeptics would say that Giffen goods exist in and only in economic text-books. We end the paper by illustrating that it is not so. Our example uses a generalized version of the multi-stage stochastic programming framework from Rasmussen and Clausen (2007) and shows that some completely rational mortgagors react to lower costs of longterm financing (reflecting a smaller market price of risk) by using more short-term financing. 2. The Markowitz model Consider a model with n risky assets with expected rate of return vector l and invertible covariance matrix R, and put 1 > = (1,...,1). The mean/variance efficient portfolios are found by solving max w w > l 1 2 cw> Rw s:t: w > 1 ¼ 1; for different values of risk-aversion c. This is a slight but convenient reparametrization of traditional formulations (e.g. Huang and Litzenberger, 1988, Chapter 3). The optimal portfolios are ^w ¼ c 1 R 1 ðl gðc; l; RÞ1Þ; where g(c;l,r) =(1 > R 1 l c)/1 > R 1 1 can be interpreted as the expected rate of return on ŵs zero-beta portfolio. A sensible definition of a Giffen good is an asset, say the ith, for which oŵ i /ol i < 0 for some c, this meaning that when the asset s expected rate of return goes up, its weight in some optimal portfolio goes down. Let us show that there are no such assets. To do this we look at the problem with the modified expected return vector l + ae i, where a 2 R and e i is the ith unit vector. The optimal portfolio in this case we can write as ^wðaþ ¼^w þ ah; where h ¼ c 1 ðr 1 e i e> i R > R 1 1 R 1 1Þ. Showing that oŵ i /ol i > 0 amounts to proving positivity of the ith coordinate of h, which we can write as! e > i h ¼ c 1 e > i R 1 e i ðe> i R 1 1Þ 2 : 1 > R 1 1 Because R 1 is strictly positive definite and symmetric, x > R 1 y defines an inner product, and strict positivity of the term in parenthesis on the right hand side of the equation above follows immediately from the Cauchy Schwartz inequality. The inclusion of a risk-free asset is handled in the same way with g replaced by the risk-free rate of return because the risk-free asset is any portfolio s zero-beta portfolio. With this result we can easily prove the Harmony Theorem from Luenberger (1998, Section 7.8) or equivalently answer the question posed in the title of Zhang (2004) that says that a newly introduced (n + 1)st asset (or project ) will be in positive demand (or: attractive ) precisely if there is strict inequality in the CAPM-like expression l nþ1 r > covðr nþ1; r M Þ ðl varðr M Þ M rþ; ð1þ where M denotes the market (or tangent) portfolio, and r s with subscripts are (stochastic) rates of returns. It is well-known, see Constantinides and Malliaris (1995, Theorem 4) but it dates back to Roll (1977), that a portfolio w is mean/variance efficient precisely if for any individual asset i we have l i r ¼ covðr i; r w Þ varðr w Þ ðl w rþ: For the portfolio ðw > M ; 0Þ> the n first necessary equations hold because the market portfolio is efficient in the old economy, and we see that the new asset is in 0-demand if equality holds in (1). Now the absence of Giffen tells us that if there is strict inequality as stated, the (n + 1)st asset has strictly positive weight in the new market portfolio. 3. The Merton model Another classic portfolio model is Merton s dynamic investment framework, see Merton (1990, Chapter 5). In its simplest case, an agent invests his wealth, W, in either a risk-free asset with rate of return r or a risky asset whose price follows a Geometric Brownian motion. Suppose the agent maximizing expected utility cares only about terminal wealth, W(T), and has a utility function with constant relative risk aversion, W ðt Þ1 c UðW ðt ÞÞ ¼ 1 c : It is optimal for this agent to invest a fixed fraction, p ¼ l r cr ; 2 of wealth in the risky asset, So if the expected rate of return of an asset (be that risky or risk-free) goes up, that asset gets higher weight in any agent s

4 R. Poulsen, K.M. Rasmussen / European Journal of Operational Research 191 (2008) portfolio. Further, by combining 2-fund separation with the Markowitz analysis from the previous section, the same conclusion is reached in a model with n rather than just one risky asset. An extension (that was actually considered in Merton s original paper; see Merton (1990, Section 5.6)) is a utility function of the form eu ðw ðt ÞÞ ¼ ðw ðt Þ WÞ1 c ; 1 c where W is some minimal required wealth; a subsistence level. Assuming initial wealth is greater than e rt W (otherwise the problem is ill-posed), the optimal strategy is to buy e rt W units of the risk-free asset and invest the rest of the wealth according to the Merton-rule from above. Thus the optimal fraction invested at time t in the risky asset is ~pðtþ ¼ W ðtþ e rðt tþ W W ðtþ so that o~pð0þ or l r cr 2 ; ¼ 1 e rt W ðt ðl rþþ1þ 1 : cr 2 W ð0þ From this we see that we can have o~pð0þ=or > 0 (for instance if e rt W =W ð0þ ¼1=2; T ¼ 30 and l r = 0.05), so the percentage of initial wealth invested in the risky asset goes up, and hence the investment in the risk-free asset goes down when the risk-free rate of return goes up. The intuition behind is that if the return of the risk-free asset goes up, you need less of it to ensure survival, and you have more money to do what you like, rather than what you have to. 4. A Mortgage choice model A way to quantify mortgage planning for many people the largest financial decisions, they ever make as a portfolio optimization problem suitable for modern OR techniques is to study minimize / ð1 cþeðxð/þþ þ c ES b ðx ð/þþ; where X(/) is the (cumulative discounted) payments from the mortgagor s dynamic portfolio strategy, /, andes b (X) =E(XjX P q b ) denotes expected shortfall (also called tail or conditional valueat-risk) based on the b-quantile q b. The minimization is done subject to a stochastic interest rate model discretized by paths through trees, each node having a universe of securities. portfolio and cash-flow constraints, transaction and mortgage origination costs as well as re-balancing constraints. This multi-stage stochastic programming problem is an extension of the models considered in Rasmussen and Clausen (2007), and it has some appealing features of both intuitive and technical natures: It takes into account both reward (low expected payments) and risk (large, extreme payments), it does so based on the coherent risk-measure (as defined by Artzner et al., 1999) expected shortfall, and it allows us to explicitly control the trade-off between risk and reward (varying c gives an efficient frontier, just like in the Markowitz model). As shown by Rockafeller and Uryasev (2000), expected shortfall gives rise to a piece-wise linear objective function. This means even large instances of the problem can be solved efficiently using standard software such as GAMS and CPLEX. For all technical details and analysis of this generalized model see Rasmussen and Zenios (2006). To model the stochastic behaviour of interest rates, we use a Vasicek model drðtþ ¼jðh rðtþþdt þ rdzðtþ; where Z is a Brownian motion. To specify the full yield curve dynamics, a market price of risk is needed. We parameterize this by k, that technically shifts the stationary mean of r to h + k under the risk-neutral measure, but more tellingly, determines the typical difference between long and short rates. This represents the fundamental trade-off in the mortgagor s problem: Short rates are typically lower than long rates, but with short-term financing, he does not know how much he will have to pay. Table 1 shows the composition of the initial optimal portfolios for two different values of the market price of risk. These two values correspond to calibration to observed Danish yield curves in October 2004 and February 2005, as depicted in Fig. 1. We first note that only the 1-year adjustable-rate bond and the 30-year callable, fixed-rate bond are

5 574 R. Poulsen, K.M. Rasmussen / European Journal of Operational Research 191 (2008) Table 1 Optimal initial loan portfolio compositions for various mortgagors facing the yield curves shown in Fig. 1 Mortgagor risk aversion(c) Optimal initial loan portfolio compositions October 2004 February 2005 Fixed rate callable Full yearly refinancing Fixed rate callable Full yearly refinancing / / / (We used the 90%-quantile for expected shortfall, 2% discounting, 1.5% transactions costs, a 7-year horizon and 6 stages). Zero coupon yield October 2004 February Maturity (years) Fig. 1. Danish yield curves from October 2004 and February 2005; full curves are calibrated model curves, dotted lines are observations. The estimated Vasicek model parameters (h,j,r) = (0.042, 0.2, 0.01) are held fixed and only the calibrated market price of risk, k, differs from October 2004 (0.017) to February 2005 (0.004). In October the difference between the 30- year and the 1-year rate is 2.8%; in February it is 1.8%. used in the optimal portfolios, although the numerical algorithm allowed for a larger universe of mortgage products (about 10 at each node). Row-wise comparisons in Table 1 give no surprises. The risk-neutral mortgagor uses full short-term financing and as risk-aversion rises more long-term financing is used. Note, however, that because short rates were historically low and the yield curve quite steep, even very risk-averse mortgagors use a significant amount (one-third to one quarter) of short-term financing. Comparing the columns tells us what a lowering of the market price of risk parameter can do to optimal portfolios. The very risk-averse mortgagor uses a larger proportion (74% compared to 68%) of long-term financing, and the risk-neutral mortgagor does not care. But for a moderately risk-averse mortgagor (c = 1/4), the lowered market price of risk, which makes short-term financing relatively less attractive, causes him to use more shortterm financing (up to 85% compared to 80% before). Although a more complicated model, the intuition is again that this mortgagor uses long-term financing initially not because he wants to, but because he has to, and lower long rates still higher than typical short rates make the necessity cheaper; like the Irish potatoes. 5. Conclusion In this note we first analyzed sensitivity to expected returns in two text-book models for optimal portfolio choice (Markowitz and Merton) and showed that the relation is as one would think; (for any asset) higher expected return raises demand (from any investor). We then demonstrated by examples the most interesting being from mortgage planning that this is not a general result. Let us end by a couple of remarks on extensions and future research. While we think our definition of a Giffen good is quite sensible, it very much takes a comparative statics view-point, that is: Differentiate the optimal solution wrt. a specific expected return parameter. One can investigate other parameter derivatives and may find surprises. But we think there is a limit to how far this analysis can be taken before running into the Lucas critique: Sensitives from a static model may tell you nothing about effects in a truly dynamic model. If you want to know how people react to a change, you must build a model where they take such changes into their optimization considerations. We see the use of stochastic programming techniques in financial engineering as very promising. The framework can be used as we did here to analyze individual mortgagors problems, but it can also be reversed to put together structured products that are optimal (in a precise quantitative sense) for investors or mortgagors. Huang et al. (2007) look at such a case and with the liberalization of capital markets enforced by new rules from the European Union much more work is needed in that direction. References Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Coherent measures of risk. Mathematical Finance 9,

6 R. Poulsen, K.M. Rasmussen / European Journal of Operational Research 191 (2008) Constantinides, G., Malliaris, A.G., Portfolio theoryhandbooks in Operations Research and Management Science: Finance, vol. 9. Elsevier. Huang, C.-F., Litzenberger, R.L., Foundations of Financial Economics. North-Holland. Huang, D., Kai, Y., Fabozzi, F.J., Fukushima, M., An optimal design of collateralized mortgage obligation with paccompanion structure using dynamic cash reserve. European Journal of Operational Research 177, Luenberger, D Investment Science, Oxford. Merton, R.C., Continuous-Time Finance. Blackwell. Rasmussen, K.M., Clausen, J., Mortgage loan portfolio optimization using multi-stage stochastic programming. Journal of Economic Dynamics and Control 31, Rasmussen, K.M., Zenios, S.A., Well ARMed and FiRM: diversification of mortgage loans for homeowners. Wharton Financial Institutions Centre Working Paper Rockafeller, R.T., Uryasev, S., Optimization of conditional value-at-risk. Journal of Risk 2, Roll, R., A critique of the asset pricing theory s tests: Part I. Journal of Financial Economics 4, Zhang, B., What kind of new asset will push up the CML?. Insurance: Mathematics and Economics